Question: Problem I
Buster Sod operates a 1200-acre irrigated farm in the Red River Valley of Arizona. Sod's principal capabilities are raising wheat, alfalfa, and beef. The Red Valley Water Authority has just given its water allotments for next year: Buster Sod was allotted 2000 acre-feet. Sod is busy preparing his production plan for next year. He figures that beef prices will hold at around $600 per ton and wheat will sell at $1.60 per bushel. He should be able to sell alfalfa at $34 per ton, but if he needs more alfalfa to feed his cattle than he can raise, he will have to pay $36 per ton to get alfalfa to his feedlot. The total alfalfa produced plus alfalfa purchased should be just enough to cover the amount used to feed the cattle plus sold to the market.
Some technological features of Sod's operation are as follows: wheat yield, 50 bushels per acre; alfalfa yield, 3 tons per acre. Other relevant information is found below:
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Activity
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Labor/ Machinery Cost
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Water Requirements (acre-ft)
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Land Requirements (acres0
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Alfalfa Requirements (tons)
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1 acre of wheat
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$8
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1.5
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1
|
|
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1 acre of alfalfa
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$30
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2.5
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1
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|
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1 ton of beef
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$40
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0.1
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0.05
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4
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Define the decision variables necessary for the LP formulation as follows:
W = wheat raised and sold (acres)
A = alfalfa raised (tons)
B = beef raised and sold (tons)
A1 =alfalfa bought (tons)
A2 = alfalfa sold (tons)
1) Formulate appropriately as an LP to maximize profit. Provide the algebraic formulation in your submission. Must be neatly presented. Integer constraints are NOT needed in this problem.
(Hint: The provided information is in many different units. The key challenge is in determining the coefficients of the objective function and constraints so that things are in compatible units, that is, compare apples to apples.)
2) Input and solve this formulation using Excel and answer the following questions.
I do not want you to rerun more than one linear program. You must explain all the steps on how you answered the question based on the output of the one standard LP output which can include Excel's sensitivity report. There is no use of SolverTable Add-In with this problem. Just use the standard Solver output. If you feel that it is not possible to answer the question with the single output, explain why this is the case.
a) How much water is being used?
b) How much beef is being produced?
c) Does Sod buy or sell alfalfa?
d) How much should Sod pay to acquire another acre of land?
e) What happens to the optimal policy if the selling price of wheat triples? Must explain how you arrived to your answer based on the given output. Do not reformulate the LP to find out the implication.
f) How much profit will Sod receive from the optimal operation of his farm?